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126 MAXIMUM LIKELIHOOD SEQUENCE DETECTION Ç start J m = m,-1 Initialize path metrics to zero m = m + 1 Loop over possible states Compute branch metrics Form candidate path metrtics Determine best candidate metric Update path metric and path history Determine best state Detect symbols using path history ortraceback end Figure 6.9 Vitorbi algorithm flow diagram. with the sum of log-likelihoods. With continuous time, we end up with an integral, giving s — arg max V J -c JV.-l r(t) — y/Es 2_] h(t — mT)q{m) dt. (6.44) As such an integral gives an estimate of noise power when the hypothesis is correct and the noise power over an infinite bandwidth is infinite, we have to assume that the noise bandwidth is finite, though at least as large as the signal bandwidth.
THE MATH 127 Expanding the square, moving the integral inside the summations, and dropping terms independent of q gives N.-l N,-l N,-l s = arg max J2 2Re{q*(m)z(m)}- J^ J^ Q*(rn 1 )q(m 2 )S{m l - m 2 ), (6.45) where p m=0 mi=(lm2=fl / oo -OO / O O h* {t - mT)r{t) dt (6.46) h*{t)h{t + er)dt. (6.47) -oc We recognize z(m) as the matched filter output for symbol s(m). The term S(£) for different values of i are referred to as the s-parameters. The double-summation term in (6.45) can be interpreted as summing the elements of a Hermitian symmetric matrix /(mi,7712) {/(τη,2,τηϊ) = /*(mi,JTi2)). The first "trick" is to rewrite the summation as the sum of the diagonal elements plus twice the real part of the sum of the lower triangular elements (due to the Hermitian property). Mathematically, JV,-1JV»-1 JV.-l ΛΤ,-lm-l Σ Σ /(ι>2)= Σ /κ)+ Σ E 2Re^( m ' fc )}· ( 6·48 ) mi— 0*712=0 m=0 m=l k=Q The second "trick" is to rewrite the sum of the lower triangular elements as sums along the off-diagonals (m — k = 1, m - k = 2, etc.), giving ΛΓ„-1 N.-l JV„-1 ΛΓ,-l m Σ Σ /("Μ>η*2) = Σ /("»·"»)+ Σ ¿2Re{/(m,m-€)}· (6-49) τηι=0τ7Ζ2=0 m=0 m=l ¿=1 Applying these two steps to (6.45) gives ΛΓ,-1 s = arg max V^ B(m), (6.50) qeSp ¿—' m=0 where branch metric B(m) is given by B(m) =Re{q*{m 2z{m) - S{0)q(m) - 2(1 - L s - 1, (6.52) where L s depends on the delay spread of the medium response as well as properties of the pulse shape p(i). For example, for root-Nyquist pulse shaping and a symbolspaced medium response (τ( — £T, I — 0, ...,L — 1), L s = L. In this case, we can
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Channel Equalization for Wireless C
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To my colleagues at Ericsson
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CONTENTS List of Figures List of Ta
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CONTENTS XI 4.2 4.3 4.4 4.5 4.6 4.1
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CONTENTS XIII 8 Practical Considera
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xvi LIST OF FIGURES 2.2 Matched fil
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XviÜ LIST OF FIGURES 6.18 Scatter
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XX Prologue Alice was nervous. Woul
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XXÜ PREFACE is introduced as neede
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ACRONYMS AC A/D ADC AMLD AMPS ASK A
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ML MLD MLPD MLSD MLSE MRC MSE MSB O
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2 INTRODUCTION Table 1.1 Possible m
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4 INTRODUCTION s o S 1 8 2 S 3 S 0
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6 INTRODUCTION A block diagram of t
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8 INTRODUCTION Figure 1.7 QPSK. tra
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10 INTRODUCTION phases (phase modul
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12 INTRODUCTION rollo« 0.22 0.8 Ί
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14 INTRODUCTION is the "channel" re
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16 INTRODUCTION 1.4 MORE MATH In th
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18 INTRODUCTION For K = 1 (order 0)
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20 INTRODUCTION The K = 4 main bloc
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22 INTRODUCTION A sample of the noi
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24 INTRODUCTION The receiver may ha
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26 INTRODUCTION 1.5.1 Reference sys
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28 INTRODUCTION a) What most likely
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CHAPTER 2 MATCHED FILTERING In this
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MORE DETAILS 33 2.2 MORE DETAILS So
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THE MATH 35 Now compare the output
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THE MATH 37 The correlation in (2.2
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THE MATH 39 correlation function f(
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THE MATH 41 Thus, in the complex pl
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THE MATH 43 to the medium response
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THE MATH 45 We can interpret f(t) a
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MORE MATH 47 10" 1 Odeg 90 deg X 18
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MORE MATH 49 With despread-level Ra
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MORE MATH 51 Figure 2.6 OFDM exampl
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MORE MATH 53 2.4.3 MF in colored no
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PROBLEMS 55 period) is examined in
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CHAPTER 3 ZERO-FORCING DECISION FEE
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MORE DETAILS 59 1/c r m —► H i
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MORE DETAILS 61 where «i = m - (d/
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MORE MATH 63 Because there is no IS
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MORE MATH 65 In the later chapter o
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PROBLEMS 67 b) What is the detected
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CHAPTER 4 LINEAR EQUALIZATION With
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THE IDEA 71 estimate of symbol s 2
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THE IDEA 73 Notice that £2 (MSE) d
Page 97 and 98: MORE DETAILS 75 The first term is t
Page 99 and 100: MORE DETAILS 77 To obtain the unity
Page 101 and 102: THE MATH 79 and SINR becomes w T h
Page 103 and 104: THE MATH 81 To obtain the MMSE solu
Page 105 and 106: THE MATH 83 where *m n = ν ^ ^ ^ '
Page 107 and 108: THE MATH 85 power is much larger th
Page 109 and 110: MORE MATH 87 IQ' 1 OC 111 ω 10 2 1
Page 111 and 112: MORE MATH 89 values of h k m (f) fr
Page 113 and 114: MORE MATH 91 where C(d(),d 0 ) . C(
Page 115 and 116: AN EXAMPLE 93 between z and s. This
Page 117 and 118: PROBLEMS 95 Another aspect of cocha
Page 119: PROBLEMS 97 The math 4.11 Consider
Page 122 and 123: 100 MMSE AND ML DECISION FEEDBACK E
Page 137 and 138: CHAPTER 6 MAXIMUM LIKELIHOOD SEQUEN
Page 139 and 140: MORE DETAILS 117 r 2 Ht, r, fcl + f
Page 141 and 142: MORE DETAILS 119 Figure 6.3 MLSD tr
Page 143 and 144: THE MATH 121 legs of the journey is
Page 145 and 146: THE MATH 123 Figure 6.8 MLSD trelli
Page 147: THE MATH 125 We would then identify
Page 151 and 152: THE MATH 129 Consider the matched f
Page 153 and 154: THE MATH 131 6.3.5.1 M-algorithm Wi
Page 155 and 156: THE MATH 133 results, DFE and MLSD
Page 157 and 158: THE MATH 135 IQ" 1 oc LU CD 10 2 10
Page 159 and 160: THE MATH 137 sometimes inaccurate w
Page 161 and 162: MORE MATH 139 POWER CONTROL effecti
Page 163 and 164: MORE MATH 141 15 POWER CONTROL 10
Page 165 and 166: MORE MATH 143 6.4.3 More approximat
Page 167 and 168: THE LITERATURE 145 EDGE is an evolu
Page 169 and 170: PROBLEMS 147 the channel to minimum
Page 171: PROBLEMS 149 c) Suppose we know «o
Page 174 and 175: 152 ADVANCED TOPICS detecting the w
Page 176 and 177: 154 ADVANCED TOPICS Table 7.2 Examp
Page 178 and 179: 156 ADVANCED TOPICS Let's look at t
Page 180 and 181: 158 ADVANCED TOPICS Thus, for MAPSD
Page 182 and 183: 160 ADVANCED TOPICS Table 7.6 Examp
Page 184 and 185: 162 ADVANCED TOPICS If we assume no
Page 186 and 187: 164 ADVANCED TOPICS Now we can focu
Page 188 and 189: 166 ADVANCED TOPICS Notice we used
Page 190 and 191: 168 ADVANCED TOPICS to achieve spat
Page 192 and 193: 170 ADVANCED TOPICS c) What are the
Page 195 and 196: CHAPTER 8 PRACTICAL CONSIDERATIONS
Page 197 and 198: MORE DETAILS 175 8.2 MORE DETAILS I
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MORE DETAILS 177 One extreme is to
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THE MATH 179 8.3.1 Time-invariant c
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THE MATH 181 With recursive channel
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MORE PRACTICAL ASPECTS 183 timing).
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THE LITERATURE 185 8.6 THE LITERATU
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PROBLEMS 187 a) Draw the new receiv
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APPENDIX A SIMULATION NOTES Perform
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FADING CHANNELS 191 where JA = A 2
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APPENDIX B NOTATION Channel Equaliz
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NOTATION 195 Variable Meaning p(t)
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198 REFERENCES [Ari98] [Ari99] [Ari
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200 REFERENCES [Bra91] W. R. Braun
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202 REFERENCES [Div98] [Dou95] [Due
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204 REFERENCES [Gon68] R. A. Gonsal
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206 REFERENCES [Kaa94] [Kai60] V.-P
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208 REFERENCES [ManOl] [Mar73] [Mar
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210 REFERENCES [Pic95] [P0088] [Pri
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212 REFERENCES [Sto93] [Sto96] [Sui
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214 REFERENCES [Wan06a] T. Wang, J.
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